Gearing



Nov. 19, 1940. R. J. s. f-"IGOTT 2,222,515

GEARING Filed Jan. 13, 194.0 2 Sheets-Sheet 1 BASIC RACK n; Too'm wm'm 73 I? a 27 7 9 3 77 25 76 f 5 25 M=TO0T+1 HEIGHT FI'ICH LINE "v PITCH LINE:

I gwumwo R- i 2% Nov. 19, 1940.

Filed Jan.

R. J. S. PIGOTT GEARING RING-6ERR 2 Sheets-Sheet 2 Patented Nov. 19, 1940 GEARING Reginald J. S. Pigott, Pittsburgh, Pa., assignor to Gulf Research & Development Company, Pittsburgh, Pa., a. corporation of Delaware Application January 13, 1940, Serial No. 313,782

7 Claims.

This invention or discovery relates to gearing; and it comprises gearing, and in particular external-internal pumping gears of one tooth difference, the teeth and tooth spaces of which correspond to those generated from a basic rack of which the contours of the teeth and of the tooth spaces are substantially segments of logarithmic spirals of formula r=ce and T=CZA respectively, wherein r and A are polar coordinates, Y is a constant equal to 0.65 or greater, Z is a constant equal to 0.85 or less, and c is a factor equal to A the tooth height; all as more fully hereinafter set forth and as claimed.

There have recently come into use gear pumps having a ring gear and a pinion, of one tooth difference, with teeth of such size and shape as to afford continuous sliding tooth contact. In rotation the teeth and inter-tooth spaces define an unbroken series of gradually expanding chambers on the inlet size of the gear combination and a series of gradually collapsing chambers on the discharge side, thereby affording a quasi-continuous pumping action. Such pumps (see my prior Patents 1,990,750, 2,053,919 and 2,055,587) have a remarkably high capacity and are advantageous in other respects. They have proved useful in many fields. In a prior Patent 1,909,117 upon which the present invention is an improvement I have disclosed and claimed a particularly advantageous tooth contour for such pumping gears. The tooth contours of the gears correspond to teeth of a basic rack, the curvature ,of which is defined by portions of logarithmic spirals. In making a set of gears according to this prior patent two cutters are used, one shaped with reference to one side of the basic rack form, so to speak, and the other formed from the other side thereof.

I have now discovered that by a modification of the tooth shapes disclosed in Patent 1,909,117 while preserving their logarithmic spiral character a basic rack can be laid out, of such nature that a single cutter made from it is capable of cutting both the externally toothed gear (pinion) and the internally toothed gear (ring gear) of a given set, with consequent simplification of the manufacturing operations. The tooth shape of the basic rack corresponds to certain logarithmic spirals. Were it not necessary to take into account clearance and backlash of the gears, the teeth and tooth spaces of the basic rack of the present invention could indeed be symmetrical about the pitch line; both could be contoured according to the same spiral. Since clearance and backlash must be provided for, the teeth and tooth spaces are not exactly symmetrical; nevertheless, one cutter formed from the basic rack serves to out both ring gears and pinions of any given ratio.

In particular, according to the invention in, its best embodiment a basic rack is laid out having teeth the curves of which are portions of logarighmic spirals of the formula r=ceand inter-tooth spaces of curvature represented by r=cewherein 1' and A are polar coordinates, s is the napierian logarithmic base, and c is a factor equal to the tooth height expressed in the same units as r (inches). These formulae hold for all basic racks for the cutting of gears of one tooth difference regardless of pitch diameter, number of teeth and tooth height. As explained in detail below, for any one set of gears the tooth height is fixed.

The above exponential constants (0.70 and 0.80) are suitable for normal requirements as regards backlash and clearance of the gear set. If greater backlash and clearance are desired, these constants can be of a greater difference, e. g. 0.68 and 0.82 respectively. The practical extremes for these constants for the two spirals are approximately 0.65 and 0.85 respectively; with a greater difierence there is apt to be excessive clearance and backlash. On the other hand, if finer fitting is desired, the two constants can have somewhat less arithmetical difference; for example, they can be 0.71 and 0.79 respectively. Thus the generalized spiral formulas for my basic rack are 1:05 and T=CEZA, wherein Y can range from about 0.85 to somewhat over 0.70, and Z can range from about 0.85 to somewhat below 0.80;

In the accompanying drawings I have shown more or less diagrannnatically examples of gearing made according to the best embodiment of the invention and charts illustrative of the principles upon which the invention is based. In the drawings,

Fig. 1 is a schematic showing of one-half of a tooth and one-half of a tooth space of a basic rack of the invention,

Fig. 2 is a chart showing a portion of one of the spirals and-illustrating how the correct portion of the spiral for tooth contours is selected,

Fig. 3 is a showing, similar to Fig. 1, of onehalf of a tooth and one-half a tooth space of an 8- tooth pinion made according to the invention,

Fig. 4 is a similar showing for a 9-tooth ring gear to mesh with the pinion of Fig. 3,

Fig. 5 is a view in elevation of a basic rack cutter,

Fig; .6 is a sectional view taken along line 6-6 of Fig. 5,

Fig. '7 is a plan view of a cutter made from the rack of Fig. and

Fig. 8 is a view in elevation of a pair of pump- 5 ing gears of tooth shape corresponding to Figs.

3 and 4.

'Referring to the drawings, Fig.1 shows in diagrammatic manner one-half of a tooth T and one-half of an inter-tooth spaces of the basic rack of the invention. The parts which are solid are conventionally indicated by hatching. The curvature of the half tooth (approximated by arcs I, 9, .23, 21) is that of a certain portion of z a logarithmic (equiangular) spiral of formula T=Cewherein r and A are polar coordinates,

, expressed in inches and in radians (1 radian:

'57.3 degrees approximately) respectively and c is /5 the tooth height expressed in inches. The equation can also be expressed as The curvature of the half tooth space (arcs 32,

36, 40) is that of a portion of a logarithmic spiral of formula Fig. 2 illustrates the manner in which the correct part of the spiral is selected for the half tooth ofFig. 1. A section of a logarithmic spiral isdrawn up according to the formula -r= e.-"

the spiral, mathematically considered can be prolonged infinitely each way from and toward 0 the pole P of the spiral. The portion of the logarect position is'shown by solid lines; the dotted" pitch line and one-half of the space width measured along the pitch line). The segments of the logarithmic spirals which subtend these right angles must be such that the radius vectors 1'1 and r2 drawn from the ends of the segments at the constant tangent angle G (tangent G=1/0.70 for the'tooth and 1/0.80 for the space) will pass through the pole. These radius vectors correspond to angles A1 and A2 as shown. The corlines show incorrect positions in that the vectors 0 drawn from the intersections i of the legs of the right angle, with the curve, at the constant tangency angle G, do not pass through the pole.

Of course, a tooth curve of any absolute size can be selected, following the above principles;

for example the tooth height M can be 0.1 inch, 1 inch, 2 inches or any other desired value.

Figs. 3 and 4 are diagrams, similar to Fig. 1, of the half-tooth and half-tooth-spaces of an 8-tooth pinion and mating 9-tooth ring gear, which are shown in elevation in Fig. 8 at H2 and H3.

Figs. 5 and 6show a basic rack embodied as a i cutter I09, which is made of suitable hardened steel with ch'amferedcutting teeth I ID as shown.

Fig. 'Ishows'a cutter made from the rack, having six cutting teeth Ill. The cutter can be made with any'number of cutting teeth greater 2,222,515 I 1 V g than two and sufficiently less than the number eccentricity (E in Fig. 8).

of teeth in the ring gear to be cut therewith, to

permit the cutter to enter the annular blank equal to inch which in turn is equal to the.

difierence in the pitch diameters or twice the In this case the width of one tooth independent of backlash would be equal to 41r/16=4.51r/18. If an attempt were made to cut a 4 inches pitch diameter, 8-tooth pinion of difierent tooth height, say inch,

in order for the tooth tips to pass in the outof-mesh position, it would be necessary for the pitch diameter of the ring gear to be 4.625 inches. In this case the tooth width would have to be changed in order to provide 9 teeth since 411'/ 16 does not equal 4.6251r/18 or conversely, the number of teeth would have to be changed in order to maintain the same tooth width.

The cutter is useful in making gear sets of any tooth ratio from 6:! up, which is a practical limit anyway for pump gear ratios. Gear sets of ratio 7:8 and 8:9 are on the whole the most useful. In these ratios the figures also indicate thenumber of teeth on the pinion and on the ring gear. Fig. 8 shows a typical gear set made according to the invention, with an 8-tooth pinion I l2 and a 9-tooth ring gear.ll3. The line of tooth contact is shown at Ii.

For the sake of illustrationthe following tabulation sets forth the constants for four different gear ratios each of inch tooth height and of pitch diameters ranging from 3 to 4.5, inch for the pinion and from 3.5 to 5 inch for the ring gear. The dimensions are in inches.

the advantages of. that described in my prior Patent 1,909,117, with the additional advantage that both gears of a given set can be cut with a single cutter formed from a single basic rack. Among these advantages are the provision of a plurality of simultaneous tooth contacts to distribute and reduce driving pressures; constant velocity ratio and good action even if the center distance .between the gears is not exactly correct; tooth contacts of, such shape and character asto form good fluid seals; and, in general, smooth andefiicient pumping action.

other metal and are made byfcuttin but they can of course be made by other procedures such.

as casting and they can be embodied in suitable non-metallic materials, v

70 The gears are ordinarily. embodied in steel or The shapes of the teeth and spaces on the basic rack, cutter, internal and external gears are, for practical purposes, most conveniently defined geometrically by a series of circular arcs, as is customary in laying out all higher order curves. The curves shown in Figs. 1, 3 and 4 are constructed in this manner.

' Fig. 1 shows the manner of laying out a basic rack for the length of one-half of a tooth and one-half of a space. Draw the parallel lines II and I2 tangent to the top of a tooth and tangent to the bottom of a space. At the right end, draw the center line I3-I4 of the tooth at right angles to the lines I I and I2. The length of the line I3I4 and of the line I6 represents the total tooth height while the line I3I1 or the line I4I8 represents one-half of the tooth length plus one-half of the space. From the point I5 on the line I3-I4 and with the radius I5--I3 draw to the left the arc I9 from the point I3 and at the final end 2I of the arc, draw the radius 20 from point I5 to 2 I. From the point 32 on the radius 20, draw the arc 23 from the point 2| to the point 25 and draw the radius 24 from the point 22 to the point 25. From the point 26 on the radius 24, draw the are 21 from the point 25 to the point 28 and draw the radius 29 from point 26 to 28. This completes the one-half of the tooth above the pitch line 43.

Beginning now at the point 30 on the center line I1I8 above line II draw therefrom with the radius 30-48 the are 32 to the right from the point I8 to the point 33. Draw the radius 34 between the points 30 and 33. From the point 35 on the radius 34, draw the are 36 from the point 33 to the point 31 and draw the radius 38 between the points 35 and 31. From the point 39 as a center on the radius 38, draw the are 40 from the point 31 to the half-way point 28 on the pitch line 43 and draw the radius 4| between the points 39 and 28. The dimensions of certain of the various radii and chords of the said arcs are set forth in the following table, the tooth height indicated by the line I3I4 or the line I1-I8 being unity.

Line I6 1.0000 Chord of arc Radius 20 .9713 32 .5757 Chord of arc Radius 38 .6337

I9 .57 95 Chord of arc Radius 24 .6745 36 .2542 Chord of arc Radius 4| .3777

23 .2056 Chord of arc Radius 29 .4402 40 .1685 Chord of arc Line I 4--I 5 .0287 21 .2018 Line 13-I1 1.5771 Radius 34 1.0040 Line I1 30 .0040

Fig. 3 shows how one-half of a tooth and onehalf of a space of an 8-tooth external pinion may be defined geometrically. The tooth contour is closely approximated as follows: Fig. 3 may be regarded as corresponding to the contour of one-half a tooth and one-half a space lying between the center line I43 of the tooth space which includes the center I and the line 44 drawn from the center I to the top of the pinion tooth, the center of the top of the tooth being indicated at 45, and the center of the bottom of the space being indicated at 46. 41 and 48 are circles tangent to the bottoms of the spaces and tops of the teeth respectively.

To draw the tooth contour for onehalf of a tooth and one-half of a space for an 8-tooth external pinion, start on the center line 44 at the point 49 below the circle 41, and with the radius 50 or 49-45 draw the arc 5| toward the right from the point to the point 52. Draw the radius from the point 49 to the pointf52. From the point 53 on the radius 50 draw the are 54 from the point 52 to the point 56 and draw the radius from the point 53 to the point 56. From the point 51 on the radius 55 draw the arc 59 from the point 56 to the point 60 on the pitch circle 8 and draw the radius 58 from the point 51 to the point 60.

From the point 14 on the center line I43 below the circle 48, draw the are 82 tothe left from the point 46 to the point 63. Draw the radius 64 from the point 14 to the point 63. From the point 65 on the radius draw the are 06 from the point 63 to the point 61 and draw the radius I61 from the point 65 to the point 61. From the point 68 on the radius I61 draw the are 69 from the point 61 to the point 60 on the pitch circle 8 and draw the radius 10 from the point 68 to the point 60. The contour of the 8-tooth external pinion is thus indicated by the tandem arcs 5|, 54, 59, 69, .66 and 62. The dimensions of thevarious radii and chords of the said arcs are set forth in the following table, the tooth height indicated by the line I43 between the circles 41 and 48 being unity.

Line 46 to circle Chord of are 69 .1434 48 1.0000 Radius 50 .9131 Radius 64 1.0542 Chord of arc 5|- .6588 Chord of are 62. .5647 Radius 55 .5484 Radius I61 .6369 Chord of are 54 .2404 Chord of are 66. .2259 Radius '58 .4002 Radius 18 .4954 Chord of are 59 .1428

Fig. 4 shows how to draw the contour of onehalf of a tooth and one-half of a space of the 9-tooth internal ring gear corresponding to the 8-tooth external pinion section shown in Fig. 3. The contour of one-half of a tooth and one-half of a space of the ring gear may be considered that shown between the center line 243 and the center line 12 drawn from the center of the gear to the point 13 at the center of the bottom of the space, the center of the top of 'the tooth being designated as point 14. The contour shown in Fig. 4 is laid out as follows, the tooth height 14 to the circle 16 or 13 to the circle 19 being unity.

From the point 15 below the intersection of the line 243 with the circumscribed circle 16 tangent to the bottom of the ring gear spaces, draw the are 11 from the point 14 to the point 18, the point 14 being in the circle 19 which is tangent to the tops of the teeth. Draw the radius from the point 15 to the point 18. From the point 8I on the radius 80, draw the are 82 from the point 18 to the point 83 and draw the radius 84, from the point 8| to the point 83. From the point 85 on the radius 84, draw the are 86 from the point 83 to the point 81 on the pitch circle I0 and draw the radius 88 from the point 85 to the point 81.

From the point 89 below the intersection of line 12 with the circle 19 which is the inscribed circle tangent to the tops of the ring gear teeth, draw the are 98 from the point 13 to the point 9I, the point 13 being at the intersection of the center line 12 with the circle 16. Draw the radius 92 between the points 89 and 9|. From the point 93 on the radius 92 draw the are 94 from the point 9| to the point 95 and draw the radius 96 between the point 93 and the point 95. From the point 91 on the radius 96 draw the are 98 from the point 95 to the point 81 on the pitch circle I0 and draw the radius 99 from the point 91 to the point 81. The contour of the ring gear in Fig. 4

,Fig. {are asfollows:

%4 v amprises the tandem arts 11'; a2,'as, .9a','941'and 90. The dimensions of the;parts referred to in Pitch line 10 between center lines 243. and

41.. 1.5771 What I claim is: v i

1. In gearing, a basic rack from which canbe generated both a ring gear-and a matingpinion of one less tooth, said rack having teeth and tooth spaces, the half-contour of which are segments of logarithmic spirals of formulas r=ce and r=ce wherein Y isa constant not less than 0.65

and Z is a constant not greater than 0.85, and wherein c is a. factor equal to /5 the tooth height.

2. In gearing, a basic rack from which can be generated both a ring gear and a mating pinion of one less tooth, said rack having teeth the halfcontour of which is a segment of a logarithmic spiral of formula r=ce-" and tooth spaces the half-contour of which is a segment of a logarith- .mic spiral of formu1a'r=cewherein c is a segment of a logarithmic spiral otiformula 1:05 wherein Z is a constant not greater than 0.85. e i I .4. In gearing, a ring gear of tooth shapecorresponding to that generated from the basic rack defined in claim 1. 5. In gearing, a pinionof tooth shape corresponding to that generated from the basic rack "defined in claim 1.

6. A ring gear and. pinion set of tooth shape corresponding to that generated from the basic rack definedin claim 1 and having at least six teeth on the pinion and one more tooth on the ring gear than on the pinion.

'7. In gearing, a multi-toothed basic rack from which can be generated both a ring gear and a mating pinion of one less tooth, said rack having teeth the half-contour of which corresponds to that segment of a logarithmic spiral of formula T cE Q which is included between a right angle whose sides are equal respectively to the tooth height measured from the pitch line and to onehalf the tooth width measured along the pitch line, and of which the radius vectors drawn from the two ends of the segment at the constant tangent angle pass through the pole of the spiral; said rack having tooth spaces the half-contour of which corresponds to that segment of a logarithmic spiral of formula 1:09- which is included between a right angle whose sides are equal respectively to the tooth space depth measured from the pitch line and to one-half the space width measured along the pitch line, and

of which the radius vectors drawn from the two ends of the segment at the constant tangent angle pass through the pole of the spiral; in which formulasc is equal to /5 the tooth height.

REGINALD J". S. PIGOTT. 

